(Todd class and Todd genus)
Let \(X\) be a smooth projective variety of dimension \(n\) over an algebraically closed field, and let \(\alpha_1, \ldots, \alpha_n\) be the Chern roots of \(T_X\). The Todd class of \(X\) is \[\operatorname{td}(X) = \prod_{i=1}^n \frac{\alpha_i}{1 - e^{-\alpha_i}} = 1 + \frac{c_1}{2} + \frac{c_1^2 + c_2}{12} + \frac{c_1 c_2}{24} + \cdots,\] with \(c_i = c_i(T_X)\). The Todd genus of \(X\) is the degree of its component \(\operatorname{td}_n(X)\) in dimension \(0\).
(Hirzebruch--Riemann--Roch for the structure sheaf)
The Todd genus of \(X\) equals \(\chi(X, \OO_X) = \sum_i (-1)^i h^i(X, \OO_X)\); in particular it is an integer.
For a curve of genus \(g\), \(\operatorname{td}_1 = c_1/2\) has degree \((2 - 2g)/2 = 1 - g = \chi(\OO_C)\). For a surface, \(c_1^2 = K_X^2\) and \(c_2 = e(X)\) is the topological Euler characteristic, and the theorem is Noether’s formula \(\chi(\OO_X) = \frac{1}{12}(K_X^2 + e(X))\).